In 2008, Borovik and Cherlin posed the problem of showing that the degree of generic transitivity of an infinite permutation group of finite Morley rank (X, G) is at most n+2 where n is the Morley rank of X. Moreover, they conjectured that the bound is only achieved (assuming transitivity) by PGL(n+1,F) acting naturally on projective n-space. We solve the problem under the two additional hypotheses that (1) (X, G) is 2-transitive, and (2) (X − {x}, Gx) has a definable quotient equivalent to (P(n−1,F), PGL(n,F)). The latter hypothesis drives the construction of the underlying projective geometry and is at the heart of an inductive approach to the main problem.